Asymmetry Amplification by a Nonadiabatic Passage through a Critical Point

TL;DR

The paper develops an integrable, dissipationless model for driving a many-body system through a second-order phase transition with weak symmetry breaking. By embedding the dynamics in a two-time Hamiltonian framework and exploiting Painlevé-II-type reductions, it shows that a small symmetry-breaking term inexorably yields a strongly asymmetric final state, with a Higgs-dominated nonadiabatic excitation spectrum. The results provide exact scaling exponents for excitation densities, linking Kibble-Zurek-type behavior to an integrable, multi-mode mean-field-like theory. The work suggests practical implications for controlled particle separation and has potential bearings on fundamental questions such as cosmological asymmetries, while outlining generalizations to higher-dimensional, integrable Hamiltonian families. Overall, the study highlights a universal mechanism by which nonadiabatic passage through a critical point amplifies asymmetry even for vanishingly small symmetry-breaking perturbations.

Abstract

We propose and solve a minimal model of dynamic passage through a second-order phase transition in the presence of symmetry breaking interactions and no dissipation. Our model generalizes the Hamiltonian dynamics of the Painleve'-2 equation to the case with many degrees of freedom, while maintaining the integrability property. The evolution eventually leads to a highly asymmetric state, no matter how weak the symmetry breaking parameter of the Hamiltonian is. This suggests a potential mechanism for strong asymmetry in the production of quasi-particles with nearly identical characteristics. The model's integrability also yields exact exponents for the scaling of the density of the nonadiabatically excited quasi-particles.

Figures (3)

• Figure 1: The potential energy $V(X_1,X_2)=-t\frac{|{\bf X}|^2}{2}+\frac{{\bf X}^4}{2}$, where ${\bf X}^2\equiv X_1^2+X_2^2$, and (a) $t=-1<0$, (b) $t=4.5<0$. Adding small nonzero $\varepsilon$ leads only to a small distortion of $V(X_1,X_2)$.
• Figure 2: The physical path, ${\cal P}$, of the evolution in the $(t,\varepsilon)$ plane goes through a critical point at $t\approx 0$ at fixed $\varepsilon\ll \beta/g$. Here, $T\rightarrow \infty$. The path ${\cal P}_{\infty}$ initially lifts $\varepsilon$ with the Hamiltonian ${\cal H'}$ to a large value $\varepsilon \gg \beta/g$ at formally infinite $|t|$. The horizontal part of ${\cal P}_{\infty}$ produces the evolution with ${\cal H}$ at large rather than small fixed $\varepsilon$. The second vertical leg of ${\cal P}_{\infty}$ brings the system back to small $\varepsilon$ at infinite $t$.
• Figure 3: (a) Trajectory $\{X_1(t),X_2(t)\}$ for $t\in [-3000,2000]$ (blue: $t<0$, red: $t>0$) and the Hamiltonian (\ref{['hamu']}); $\varepsilon=0.001$, $e_1=-1/2$, $e_2=1/2$, $X_1(t=-3000)=X_2(t=-3000)=-0.1$, $P_1(t=-3000)=-20$, $P_2(t=-3000)=20$. (b) Coordinates $X_1(t)$ (black) and $X_2(t)$ (purple) as functions of time for the same $\varepsilon$, $e_{1,2}$, and initial conditions. After the phase transition at $t=0$, both $X_1$ and $X_2$ pick up the amplitude with initially equal rate but for longer times, $t>1/\varepsilon$, $X_1(t)$ keeps growing, while $X_2(t)$ oscillates near $\langle X_2 \rangle =0$. (c) Numerical check for $\varepsilon$-independent behaviour of $\Delta I_{1,2}$ at $I^{-\infty}_{1,2}=0.0001$. The inset resolves the interval $\varepsilon \in [0.2,1.8]$. The time interval for simulations is $t\in (-500,3000)$ and the time-step is $dt=0.0005$. Each point is the average of the final adiabatic invariant over $6000$ trajectories with different initial angles, in the action-angle variables, taken from a uniform distribution $\varphi_{1,2} \in (0,2\pi)$. (d) Numerical confirmation for Eq. (\ref{['corrected']}): the logarithmic dependence of $\Delta I_1$ on $I_{1,2}^{-\infty}= I^{-\infty}$ and a constant value for $\Delta I_2$. Here, $\varepsilon=1$ and the other parameters are as in (c).